article

Title: $C^*$-Algebras associated with complex dynamical systems

Authors: Tsuyoshi Kajiwara and Yasuo Watatani

Issue: Volume 54 (2005), Issue 3, 755-778

Abstract:

$Iteration of a rational function $R$ gives a complex dynamical system on the Riemann sphere. We introduce a $C^{*}$-algebra $\mathcal{O}_R$ associated with $R$ as a Cuntz-Pimsner algebra of a Hilbert bimodule over the algebra $A=C(J_R)$ of continuous functions on the Julia set $J_R$ of $R$. The algebra $\mathcal{O}_R$ is a certain analog of the crossed product by a boundary action. We show that if the degree of $R$ is at least two, then $C^{*}$-algebra $\mathcal{O}_R$ is simple and purely infinite. For example if $R(z)=z^2-2$, then the Julia set $J_R=[-2,2]$ and the restriction $R:J_R\to J_R$ is topologically conjugate to the tent map on $[0,1]$. The algebra $\mathcal{O}_{z^2-2}$ is isomorphic to the Cuntz algebra $\mathcal{O}_{\infty}$. We also show that the Lyubich measure associated with $R$ gives a unique KMS state on the $C^{*}$-algebra $\mathcal{O}_R$ for the gauge action at inverse temperature $\log(\deg R)$, if the Julia set contains no critical points.$